The S₃-symmetric tridiagonal algebra

The tridiagonal algebra is defined by two generators and two relations, called the tridiagonal relations. Special cases of the tridiagonal algebra include the q-Onsager algebra, the positive part of the q-deformed enveloping algebra Uq(sl₂), and the enveloping algebra of the Onsager Lie algebra. In this paper, we introduce the S₃-symmetric tridiagonal algebra. This algebra has six generators. The generators can be identified with the vertices of a regular hexagon, such that nonadjacent generators commute and adjacent generators satisfy a pair of tridiagonal relations. For a Q-polynomial distance-regular graph Γ we turn the tensor power V⊗ 3 of the standard module V into a module for an S₃-symmetric tridiagonal algebra. We investigate in detail the case in which Γ is a Hamming graph. We give some conjectures and open problems.